We study the quantum dynamics of a single-Cooper-pair box biased by a classical voltage and also irradiated by a single-mode quantized field.We demonstrate that under weak damping of the quantized field,the collapse-revival phenomena can exist in this system,and the oscillations of the collapse and revival depend sensitively on the initial state of the single-mode quantized field and the damping rate κ.We also demonstrate that this system can show the beats phenomena.